Indoors on 1.5x, like for basketball,volleyball,even wrestling and gymnastics, a 70-200 sucks...it's too damned LONG on a crop--frame body.

This issue with Sigma has long,long,long been poor autofocusing on some examples of their lenses...with some camera models.

When the camera makers update their AF protocols, the third-party lens makers have to reverse-engineer their lenses for the newer cameras; this is usually a fairly uncommon, rare occurrence, but say like, when the D200 came out and consumer Nikon's got back-button focus, Sigma's older lenses were suddenly, in many cases, not working right. Honestly...I got rid of my two big, high-end late 1990's Sigmas because my 2005-and-newer Nikons didn't focus "right" with them...wayyyy too many instances of severe focus hunts with the HSM EX-series 180/3.5 APO-Macro and the EX-series HSM 100-300mm f/4.

I LOVED the Nikon 50-135mm f/3.5 Ai-S Nikkor manual focus on the D1,D1h,and D2x crop-frame d-slrs...the 50-135mm range was SUPER-handy!!! There is simply no comparison between the handiness of a 50-135 and a 70-200 or 80-200 on a crop-frame body; the shorter lens makes a HUGE difference in usefulness. So,so,soooooo much better for parades, events, and portraits/scenics, with the 50mm versus 70mm short end.

Now that we've gone up to 24-MP on APS-C...there's less need for the 200mm top end all of the time, but for longer-range stuff, 200mm is better than 135mm was, and much longer than 100mm is.

... The amount of light coming into the lens and hitting the sensor will be the same on both full frame and crop at the same aperture but on crop the depth of field won't be as narrow as on full frame at the same aperture. To find out what aperture it's equivalent to you have to multiply the aperture by the crop factor 1.6 for canon to find what the depth of field will be. The light coming in will remain the same.

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As I said: Crop factor has nothing to do with aperture. While there is a difference in DoF, it relates to the size of the sensor, and is not a straight "1.6 greater".

I have the 70-200 f2.8 and it was absolutely worth it. I also got the 2x teleconverter and while there are compromises when using a teleconverter it was also worth it when considering the alternatives. In fact when I compare these two together, they were about $2500, I see this combo as the best value in Canon white glass for a very broad mission profile.

Of course it really depends on your mission but mine is so varied that if I take this setup along with my 24-105 f4 I'm ready for a wide diversity of situations.

I shoot with a 6D Mark II. I will get the 7D Mark III when it becomes available. I hear really good reports about the 80D and would expect the 70-200 f2.8 and 2x teleconverter combo to work well on it. I really like the 6D Mii for its high ISO performance: think indoor sports.

I have the 70-200 f2.8 and it was absolutely worth it. I also got the 2x teleconverter and while there are compromises when using a teleconverter it was also worth it when considering the alternatives. In fact when I compare these two together, they were about $2500, I see this combo as the best value in Canon white glass for a very broad mission profile.

Of course it really depends on your mission but mine is so varied that if I take this setup along with my 24-105 f4 I'm ready for a wide diversity of situations.

I shoot with a 6D Mark II. I will get the 7D Mark III when it becomes available. I hear really good reports about the 80D and would expect the 70-200 f2.8 and 2x teleconverter combo to work well on it. I really like the 6D Mii for its high ISO performance: think indoor sports.

sorta rich for my blood. 70 on a crop sensor is too long for indoors. I have a Sigma 28-105 left over from film shooting. It's a 4 - 5.6 so not great but a pretty useful focal length. Something around 24-135 at 2.8 would be the cats meow!

I got the f/4 version of the Nikon 70-200.
The f/4 lens is half the weight of the f/2.8 lens.
I would have loved the faster f/2.8 lens, but it is too heavy for this old man to handle for a long shoot.

Having now shot with the f/4 lens, I really do not want the weight of the f/2.8 lens, except on a monopod.

Except for certain static shooting where I can use a monopod, I prefer to be more mobile with my gear. I find it easier to pan with the action, like shooting a shotgun, rather than pivoting around a monopod.

Agree. It would be a great DX/crop lens for field sports (football, soccer and lacrosse), shooting on the sidelines. But it would be too short for shooting from the bleachers.
For gym sports (basketball and volleyball), it should do just fine on the floor or the bleachers.

But it is going to be a big heavy lens.
The closest comparable lens is the Nikon 24-120 f/4. So think of this lens, even bigger in diameter and heavier.

Having now shot with the f/4 lens, I really do not want the weight of the f/2.8 lens, except on a monopod.

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Weight and size are certainly important considerations. I normally do not travel by air with the 2.8 70-200 because of this and the risk of theft and damage to an expensive lens. I have an EF 70-300 f 4-5.6 for travel for many of the reasons you pointed out and it is 1/4 the cost.